Negative Exponents

Learn to work with negative exponents and reciprocals. Math 10-C Alberta mathematics.

Lesson 2.5 of Number Systems & Operations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• What Negative Exponents Mean. Discover why a negative exponent means 'take the reciprocal' by applying the product rule to show that a base with a negative exponent equals one over that base with a positive exponent.
• Evaluating Expressions with Negative Exponents. Work through examples of evaluating negative exponents — including fractions raised to negative powers — and learn the two-step process for combining negative and fractional exponents.
• Real-World Application. See negative and fractional exponents used together in a paleontology formula that estimates how fast a dinosaur could run from its stride length and foot length.

Inside the lesson — a free preview

Reciprocals — A Quick Review

What Is a Reciprocal?. Two numbers are reciprocals if their product equals 1.

4 × (1)/(4) = 1, so 4 and (1)/(4) are reciprocals.

(2)/(3) × (3)/(2) = 1, so (2)/(3) and (3)/(2) are reciprocals.

To find the reciprocal of any number, flip it: the reciprocal of (a)/(b) is (b)/(a). The reciprocal of a whole number like 7 is (1)/(7).

Discovering Negative Exponents

Using the Product Rule. Let's use what we already know about exponent rules to figure out what a negative exponent means. The product rule says:

a^m · a^n = a^(m+n)

Apply this to 5^(-2) · 5^2:

5^(-2) · 5^2 = 5^(-2+2) = 5^0 = 1

So 5^(-2) multiplied by 5^2 equals 1. That means 5^(-2) and 5^2 are reciprocals.

Since 5^2 = 25, and the reciprocal of 25 is (1)/(25):

5^(-2) = (1)/(5^2) = (1)/(25)

The Rule

What the Rule Means. A negative exponent means "take the reciprocal." The base moves from the numerator to the denominator (or vice versa), and the exponent becomes positive.

This also works in reverse:

(1)/(x^n) = x^(-n)

And it works with fractions — a negative exponent on a fraction flips the fraction:

((a)/(b))^(-n) = ((b)/(a))^(n)

Common Misconception. A negative exponent does NOT make the result negative. It makes a reciprocal.

For example: 2^(-3) = (1)/(8) (positive!), not -8.

Worked example: Basic Examples

Example 1 — Evaluate 7^(-2). Apply the rule — move the base to the denominator and make the exponent positive:

Step 1: Apply the negative exponent rule:

7^(-2) = (1)/(7^2)

Step 2: Evaluate the positive exponent:

Answer: (1)/(49)

Example 2 — Evaluate ((10)/(3))^(-3). A negative exponent on a fraction flips the fraction:

Step 1: Flip the fraction to handle the negative exponent:

((10)/(3))^(-3) = ((3)/(10))^(3)

Step 2: Evaluate with the positive exponent:

= (3^3)/(10^3)

= (27)/(1000)

Answer: 0.027

Example 3 — Evaluate (-(3)/(2))^(-3). Step 1: Flip the fraction first:

(-(3)/(2))^(-3) = (-(2)/(3))^(3)

Step 2: Cube everything, including the negative sign:

= ((-2)^3)/(3^3)

= (-8)/(27)

Answer: -(8)/(27) ≈ -0.296

Watch the Sign!. Don't lose the negative sign on the base! When the base is negative, pay close attention.

(-2)^3 = -8 because a negative number raised to an odd power stays negative.

(-2)^2 = 4 because a negative number raised to an even power becomes positive.

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